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Optical Bloch Equations

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Atoms, Molecules and Optical Physics 2

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

To set the stage, in Sect. 10.1 we take a look onto an experiment from modern quantum optics. In Sect. 10.2 the important “dressed state” model is introduced to analyze the two level system in a quasi-monochromatic light. Section 10.3 presents several characteristic experiments which may be explained effortless with this model. Section 10.4 derives the theoretical framework for treating such systems quantitatively, starting with the fundamental Liouville-von-Neumann equation from which the “Optical Bloch equations” are derived. In Sect. 10.5 we apply these to a number of important questions which in earlier chapters could only be discussed by hand waving arguments. On these grounds, Sect. 10.6 develops some basics of short pulse spectroscopy. Finally, Sect. 10.7 introduces a somewhat more complex application, the STIRAP method, today of increasing interest also in the context of “quantum information”.

Up to now we have treated optically induced processes exclusively in the framework of perturbation theory. If, however, the probability densities of the excited states become comparable to those of the initial states this is no longer sufficient. Also, perturbation theory does not answer the question about the type of radiation which is re-emitted in such a case. By spontaneous emission, inevitably, mixed states are created. To include these into a formal description we have to apply the concepts developed in Chap.  9 . These and related questions are at the heart of quantum optics and subject to the present chapter.

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Notes

  1. 1.

    Typically, a two level system may be realized for j b =j a +1 with maximum or minimum projection quantum numbers by the states |γ a j a ±j a 〉 and |γ b j b ±j b 〉 with circularly polarized light, q=±1, for excitation. A classical example is the 3 2S1/2 F=2 M F =2↔3 2P3/2 F=3 M F =3 transition, populated by optical pumping (see Appendix D).

  2. 2.

    However, when studying processes of higher order (e.g. multi-photon processes), the application of the RWA is no longer trivial.

  3. 3.

    This will be different as soon as another photon is involved which is absorbed or emitted.

  4. 4.

    The phase factors −i can be traced back to our choice of the phase in the definition (2.87) of the electric field.

  5. 5.

    If the intensity is sufficiently low, so that optical pumping can be neglected, these expressions may also be applied to transitions with several sublevels involved. One just has to replace the factor 3 in (10.45) by g b /g a (see footnote 2 in Chap. 5, Vol. 1).

  6. 6.

    These initial conditions imply again that the radiation field is switched on at t=0.

  7. 7.

    Note that due to our definition (1.35) for the electric field of the radiation the off diagonal matrix elements differ from those of Bergmann et al. (1998) by an (insignificant) phase factor ±i.

  8. 8.

    We use here, different from Bergmann et al. (1998), the standard terminology according to NIST.

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Authors and Affiliations

  1. Max-Born-Institut für Nichtlineare Optikund Kurzzeitspektroskopie, Berlin, Germany

    Ingolf V. Hertel & Claus-Peter Schulz

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  1. Ingolf V. Hertel
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Acronyms and Terminology

c.c.:

‘complex conjugate’.

CW:

‘Continuous wave’, (as opposed to pulsed) light beam, laser radiation etc.

E1:

‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation.

EPR:

‘Electron paramagnetic resonance’, spectroscopy, also called electron spin resonance ESR (see Sect. 9.5.2 in Vol. 1).

FORT

‘Far-off-resonance optical dipole trap’, for trapping single atoms; a typical setup is shown in Fig. 10.1.

FPI:

‘Fabry-Pérot interferometer’, for high precision spectroscopy and laser resonators (see Sect. 6.1.2 in Vol. 1).

FWHM:

‘Full width at half maximum’.

HBT:

‘Hanbury Brown and Twiss’, experiment, to determine the lateral correlation of light by a second-order interferometric measurement (see Sect. 2.1.6).

MOT:

‘Magneto optical trap’, for a typical setup see e.g. Fig. 6.26.

NIST:

‘National institute of standards and technology’, located at Gaithersburg (MD) and Boulder (CO), USA. http://www.nist.gov/index.html.

NMR:

‘Nuclear magnetic resonance’, spectroscopy, a rather universal spectroscopic method for identifying molecules (see Sect. 9.5.3 in Vol. 1).

ODE:

‘Ordinary differential equation’.

RWA:

‘Rotating wave approximation’, allows to solve the coupled equations for a two level system in a strong electromagnetic field in closed analytical form (see Sect. 10.2.3).

STIRAP:

‘Stimulated Raman adiabatic passage’, special type of optical pumping, see Sect. 10.7.

SVE:

‘Slowly varying envelope’, approximation for electromagnetic waves (see Sect. 1.2.1, specifically Eq. (1.38)).

UV:

‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between 100 nm and 400 nm according to ISO 21348 (2007).

VUV:

‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between 10 nm and 200 nm according to ISO 21348 (2007).

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Hertel, I.V., Schulz, CP. (2015). Optical Bloch Equations. In: Atoms, Molecules and Optical Physics 2. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54313-5_10

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